#### Abstract

This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer lattice . The dynamics of each given cell depends on itself and its nearest left or right neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Our technique is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we construct a suitable wave profiles set and derive the existence of traveling wave solutions by using Schauder's fixed point theorem.

#### 1. Introduction

Recently, there have been an increasing activity and interest in the study of nervous systems, namely, in the study of equations modeling neural networks, which are applied to a broad scope of fields such as image and video signal processing, robotic and biological versions, and higher brain functions (see [1–6] for more details). The methodology of cellular neural networks (for short, CNN) was first proposed by Chua and Yang [7–9] as an achievable alternative to fully connected neural networks in electric circuit systems. The dynamic evolution of the network is governed by the assumed dynamics of the individual processing units and their reciprocal interactions. The infinite system of ordinary differential equations for CNN distributed in a one-dimensional integer lattice can be described by where is the piecewise-linear output function given by The quantity is called a threshold or bias term and is related to independent voltage sources in electric circuits. The real constant coefficients , , and of the output function constitute the so-called space-invariant template that measures the synaptic weights of self-feedback and neighborhood interaction. In recent years, when the output function is defined as (2), some incisive mathematical analyses have subsequently been done; see [1–3, 7–9] and the references cited therein.

However, to be more realistic, the neural models should be incorporated into time delays, since the transmission of information from one neuron to another is not instantaneous. In models of electronic neural networks, the dynamics of each given cell depends on itself and its nearest left or right neighbors where delays exist in left or right neighborhood interactions due to, for example, finite switching speed and finite velocity of signal transmission (see [7]). Such assumptions can be also found in [1, 10–13]. In the past decades, there are many mathematical results on the study of dynamics for neural models. Particularly, the investigation of traveling wave solutions has attracted much attention due to its significant nature in biology, chemistry, epidemiology, and physics, such as reaction-diffusion equations [14–23] and lattice equations [2, 8, 24, 25]. For delayed CNN models (for short, DCNN) with the piecewise-linear output function (2), the existence of traveling waves has been investigated in [1, 11–13]. If the output function is nonlinear, for example, or the authors of [10] also derived the existence of traveling wave solutions. Recently, when the propagation of signal is very slow, authors of [26] considered CNN models with infinite time delays. More precisely, Yu et al. [26] investigated the existence and nonexistence of monotonic traveling waves for the following more general CNN models with infinite distributed time delay terms: for , . Here , , and are nonnegative constants; may be a positive constant or infinity; and the output function is nondecreasing. Here the dynamics of each given cell depends on itself and its nearest left or right neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Moreover, we point out that the CNN models with discrete time delays or no delays case can be included in this model by choosing suitable kernel functions (see [26]).

Note that some known results on the existence of traveling waves for (5) can be obtained only when the output functions are nondecreasing. But the output functions may not be nondecreasing. For example, let us consider the following two nonmonotonic output functions: It is obvious that the previous results in [2, 3, 10, 12, 27] cannot be applied for CNN or DCNN models with nonmonotonic output functions like (6) or (7). Therefore, it gives us the motivation to consider the existence of traveling wave solutions for DCNN model (5) with nonmonotonic output functions.

Throughout this paper, we assume that there exists a such that and satisfy the following assumptions.(F1) is an odd function, , and there exist with and such that where , , , and , (F2), for , and for .(F3)There exists such that (F4) for and for .(J) are piecewise continuous functions satisfying where . If , we further assume that

*Remark 1. *If satisfies the following hypotheses,(H1) is a continuous odd function, , for , , , and there exists such that for ,(H2) for ; in addition, exists and for ,then it is obvious that (F1)–(F3) hold and 0, , and are three equilibria of (5).

*Remark 2. *If , then the nonmonotonic output functions (6) and (7) satisfy assumptions (F1)–(F4).

Our purpose is to investigate the existence of traveling waves for (5) with nonmonotonic output functions. A traveling wave solution of (5) is a special translation invariant solution of (12) with the form , , and for a wave profile , , with a given wave speed . Letting , it follows that must be a solution of the following wave profile equation: Under assumptions (F1)-(F2), it is easily seen that and are three equilibria of (13). We are mainly interested in the existence of traveling waves satisfying one of the following asymptotic boundary conditions:

Since the output functions are nonmonotonic, our main idea is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we can apply the results in [26] and Schauder’s fixed point theorem to derive the existence of traveling wave solutions. Now we state our main results.

Theorem 3. *Assume (F1)–(F3) and (J) hold. If and , then there exists such that the following statements hold.*(1)*For each , (5) has a traveling wave with speed such that and satisfies (BC1). Moreover,
**where is the smallest positive root of characteristic function of (13) at 0. If, in addition, (F4) holds, then .*(2)*For each , (5) has a traveling wave with speed such that and satisfies (BC3). Moreover,
**If, in addition, (F4) holds, then .*

Theorem 4. *Assume (F1)–(F3) and (J) hold. If and , then there exists such that the following statements hold.*(1)*For each , (5) has a traveling wave with speed such that and satisfies (BC2). Moreover,
where is the largest negative root of characteristic function of (13) at 0. If, in addition, (F4) holds, then .*(2)*For each , (5) has a traveling wave with speed such that and satisfies (BC4). Moreover,
**If, in addition, (F4) holds, then .*

Theorem 5. *Assume (F1)–(F3) and (J) hold. If and , then the following statements hold.*(1)*For any , (5) has a traveling wave with speed such that and satisfies (BC3). Moreover,
**where is a unique real root of characteristic function of (13) at 0. If, in addition, (F4) holds, then . *(2)*For any , (5) has a traveling wave with speed such that and satisfies (BC3). Moreover,
**If, in addition, (F4) holds, then .*

Theorem 6. *Assume (F1)–(F3) and (J) hold. If and , then the following statements hold.*(1)*For any , (5) has a traveling wave with speed such that and satisfies (BC2). Moreover,
**where is a unique real root of characteristic function of (13) at 0. If, in addition, (F4) holds, then . *(2)*For any , (5) has a traveling wave with speed such that satisfies (BC4). Moreover,
**If, in addition, (F4) holds, then .*

*Remark 7. *In the above theorems, we obtain that the traveling waves either converge to the nontrivial equilibrium ( or ) or oscillate on the nontrivial equilibrium at infinity. Furthermore, we give a sufficient condition for the convergence of traveling waves to the nontrivial equilibrium. But under what conditions the wave will oscillate on the nontrivial equilibrium at infinity is an interesting problem. On the other hand, similarly to the proofs of nonexistence of traveling waves in [26], we can also obtain the same conclusions for nonmonotonic DCNN models.

*Remark 8. *The different results can be obtained through the choice of the signs for and and (5) reduces to kinds of CNN model with mixed delays by choosing suitable kernel functions. Please refer to [26].

The remainder of this paper is organized as follows. In Section 2, we recall some properties of the characteristic function of (13). Then, in Section 3, we construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Based on the construction of two nondecreasing functions, we devote Section 4 to proofs of the existence of traveling waves for (5). Our approach is to squeeze the nonmonotonic output functions and Schauder’s fixed point theorem in a suitable Banach space. Furthermore, according to the construction of a wave profiles set, we can obtain the exponential asymptotic behavior of traveling waves in the infinity.

#### 2. Properties of the Characteristic Function

In this section, we recall some properties of the characteristic function for (13). The characteristic function arises from the linearized equation of (13) at the equilibrium solution and is given by The main properties of the characteristic function are stated in the following lemmas, and all the proofs can be found in our recent work [26].

Lemma 9. *Assume that and . Then there exists a unique such that *(1)*if , then there exist two positive numbers and with such that ;*(2)*if , then , and if , then ,
*(3)*if , then for all .*

Lemma 10. *Assume that and . Then there exists a unique such that *(1)*if , then there exist and with ;*(2)*if , then , and if , then ,
*(3)*if , then for all .*

Lemma 11. *Assume that and . Then we have the following:*(1)*for any , there exists a unique real root such that . Moreover, for , there exists such that for and for ;*(2)*for , for all .*

Lemma 12. *Assume that and . Then we have the following:*(1)*for any , there exists a unique real root such that . Moreover, for any , there exists such that for and for ;*(2)*for , for all .*

According to the results of the above lemmas, in Section 4, we can see that the roots of characteristic equation play crucial roles in studying the behavior solutions of (13) near the equilibrium 0.

#### 3. Construction of Nondecreasing Functions

In this section, we will construct two nondecreasing functions such that lies between the two functions. Based on the construction of the nondecreasing functions, we can apply the results in [26] to derive the existence results.

Let us define functions and by Then and satisfy the following properties.

Lemma 13. *Assume that (F1)–(F3) hold.*(1)* and are nondecreasing and Lipschitz continuous on .*(2)* for all .*(3)* and for all .*(4)* and there exists an with such that . Moreover, we have for any and for any .*(5)*There exists such that
*

*Proof. *It is obvious that the assertions of , , and hold. Therefore, we only prove the assertions of parts and .

(2) By the definition of and (F2), we have for . According to (F1)-(F2), it is obvious that and for . Thus, for .

(4) Obviously, . Since , it follows that and there exists such that . By the assertion of part , we have
Thus, there exists an such that
On the other hand, since and for any , by the definition of , we have for any . This completes the proof.

#### 4. Existence of Traveling Wave Solutions

Now, applying the techniques developed in [26] to the functions and , we prove the main theorems in the sequel.

##### 4.1. Traveling Waves with (BC1) and (BC3)

First, we consider the existence of traveling wave solutions satisfying condition (BC1). Once there exist traveling waves with (BC1), then we can easily obtain that (5) admits traveling wave solutions satisfying condition (BC3).

Let us define the operator by where is a constant and According to (F1) and , is well defined. It is easy to see that a fixed point of or a solution of the equation is a traveling wave solution of (5).

Let be defined as in (30) with replaced by ; that is, where According to Lemma 13, it is easily seen that and are nondecreasing on and

Now we define the functions and by Since , it is obvious that . Similar to the proof of Lemma 3.1 in [26], it is easily seen that the following lemma holds.

Lemma 14. *Assume that (F1)–(F3) hold. Then, for any ,
*

Since is nondecreasing and satisfies the assumptions illustrated in [26], then we obtain the following result.

Lemma 15. *Assume that (F1)–(F3) hold. Then, for any , there exists a fixed point of such that and for all . Moreover,
*

*Proof. *The proof is the same as that of Theorem 1.1 of [26], and it is omitted.

By Lemmas 13–15 and (33), we easily verify that For a given number , let It is easy to see that is a Banach space. Then we define the following set: where . By (39) and the definitions of , , we easily check that is a nonempty closed convex subset of . Thus, we have the following assertion.

Lemma 16. *Assume that (F1)-(F2) hold. Then we have*(1)*;*(2)* is completely continuous with respect to the norm in .*

*Proof. *(1) For , by (39) and Lemmas 14–15, it follows that
which implies that .

(2) For any , it follows from (F1) that
Thus, we have
which implies that is continuous. On the other hand, for any , , we have , and
for , . Since , it follows that the family of functions is uniformly bounded and equicontinuous in . Thus, by Arzela-Ascoli theorem, for any given sequence in , there exist and such that uniformly for in any compact subset of . Since for any , we have for any , and therefore .

Next, we claim that
In fact, since and , we have
Since and , it follows that
Thus, it is easy to see that is compact with respect to the norm in . The proof is complete.

Now we start the proofs of Theorems 3 and 5.

*Proof of Theorem 3. *(1) By Lemma 3.4, Schauder’s fixed point theorem implies that there exists such that . Since
we can easily obtain that, for ,
Moreover, we have
Next, we prove that
In fact, if , then exists. Taking and applying L’Hospital’s rule to (32), we can obtain which yields .

Now let us consider . Since it is impossible that is eventually monotone, is oscillating on positive infinity and then there exist two sequences with as and with as such that
Since is continuous, for , there exists a such that
For such a , there exists a large enough number such that
If , we choose a large enough integer such that for all . Thus, for , we have
If , for the above , there exists a sufficiently large number such that
Now let us choose a large enough integer such that for all . Therefore, for , we have

According to the above argument and taking the limit as , we obtain
Since is arbitrary, it follows that
Similarly, we can obtain
If , then (61) and (F2) imply that
which leads to a contradiction. On the other hand, if , then (60) and (F2) also imply that
which also gives a contradiction. Hence we conclude that .

If, in addition, (F4) holds, then we claim that . Suppose the claim is false; that is, . Let such that
Then we consider the following three cases.*Case *1. .

If , according to (60), we have since . Therefore . Then, according to (60) and , we have , which leads to a contradiction.*Case *2. .

Similar to the discussion of Case 1, the case cannot happen.*Case *3. .

By (60)-(61) and (F4), it follows that
which is impossible.

Thus, ; that is, the limit exists. Taking and applying L’Hospital’s rule to (32), we obtain that . Hence, the assertion of part (i) follows.

(2) Since is an odd function, letting , then (13) reduces to
It is easy to check that (66) has the same form as (13) and hence we can obtain the result of part (ii). The proof is complete.

*Proof of Theorem 5. *Let us assume that and . Similar to the definitions of (36), for any , we define the continuous functions as follows:
where is defined in Lemma 11. Similar to the proofs of Lemmas 14–16, it is easily seen that and is compact with respect to the norm in , where . Hence, by the same arguments as those of the proof of Theorem 3, we can obtain the assertions of the theorem. This completes the proof.

##### 4.2. Traveling Waves with (BC2) and (BC4)

By arguments similar to those of the proof of Theorem 3, in this section we show the existence of traveling wave solutions satisfying conditions (BC2) and (BC4).

For , we define the operator by where According to (F1) and (J), is well defined. It is obvious that a fixed point of is a traveling wave solution of (5). Similar to (33), we may define through in the same way.

First, we consider and . Let us define the continuous functions by Since is nondecreasing and satisfies the assumptions illustrated in [26], similar to Theorem 1.2 of [26], we can obtain the following results.

Lemma 17. *Assume that (F1)–(F3) hold. Then, for any , one has
*

Lemma 18. *Assume that (F1)–(F3) hold. Then, for any , there exists a fixed point of such that and for all . Moreover,
*

Now, fixing a number , we define the set by It is easy to see that is a nonempty, closed, and convex subset of . Similar to the proof of Lemma 16, we have the following lemma.

Lemma 19. *Assume and . The following assertions hold.*(1)*;*(2)* is compact with respect to the norm in .*

Now we start the proofs of Theorems 4 and 6.

*Proof of Theorem 4. *According to Lemma 19 and Schauder’s fixed point theorem, there exists such that . Since for and , we can easily obtain that, for ,
By the same arguments as those of the proof of Theorem 3, we obtain the assertions of the theorem. This completes the proof.

Next, we consider the case and . For any , let us redefine the continuous functions by Then, similar to that of Theorem 5, we obtain the conclusion of Theorem 6.

#### 5. Conclusion

In this paper, the motivation of our work is to consider the existence of traveling wave solutions for DCNN model with nonmonotonic output functions. We have developed a technique to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. By adopting Schauder’s fixed point theorem, the existence of traveling wave solutions has been derived.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

Zhi-Xian Yu is partially supported by the National Natural Science Foundation of China, Shanghai Leading Academic Discipline Project (no. XTKX2012), Innovation Program of Shanghai Municipal Education Commission (no. 14YZ096), and Hujiang Foundation of China (B14005). Rong Yuan and Ming-Shu Peng are partially supported by the National Natural Science Foundation of China and RFDP. Cheng-Hsiung Hsu is partially supported by the National Science Council and NCTS of Taiwan.